My question is, will deleting saves also delete achievements and unlockables from the new DLCs Witch Hunt, Lelianas Song and Golems Of Amgarrak?

No, it will not. You'll still have your achievements earned, regardless of whether you deleted your save game, or not. Now, the only way possible to get rid of all your unlockables, and achievements is by deleting your BioWare folder located in your Documents (which is something that I accidentally did once ).